A smoothed GPY sieve

Motohashi, Yoichi; Pintz, János

doi:10.1112/blms/bdn023

Cited by 21 publications

(26 citation statements)

References 5 publications

(16 reference statements)

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“…Theorems 22,24,26, and 28 use the asymptotics established in Theorems 19 and 20, in combination with Lemma 18, to give various criteria for bounding H m , which all involve finding sufficiently strong candidates for a variety of multidimensional variational problems; these theorems are proven in the 'Reduction to a variational problem' section. These variational problems are analysed in the asymptotic regime of large k in the 'Asymptotic analysis' section, and for small and medium k in the 'The case of small and medium dimension' section, with the results collected in Theorems 23,25,27,and 29. Combining these results with the previous propositions gives Theorem 16, which, when combined with the bounds on narrow admissible tuples in Theorem 17 that are established in the 'Narrow admissible tuples' http://www.resmathsci.com/content/1/1/12 section, will give Theorem 4.…”

Section: Organization Of the Papermentioning

confidence: 99%

“…While such an estimate remains unproven, it was observed by Motohashi and Pintz [27] and by Zhang [3] that a certain weakened version of EH [ϑ] would still suffice for this purpose. More precisely (and following the notation of our previous paper), let , δ > 0 be fixed, and let MPZ[ , δ] be the following claim: …”

Section: Distribution Estimates On Arithmetic Functionsmentioning

confidence: 99%

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Variants of the Selberg sieve, and bounded intervals containing many primes

Polymath¹

2014

Res Math Sci

130

172

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For any m ≥ 1, let H m denote the quantity lim inf n→∞ (p n+m − p n ). A celebrated recent result of Zhang showed the finiteness of H 1 , with the explicit bound H 1 ≤ 70, 000, 000. This was then improved by us (the Polymath8 project) to H 1 ≤ 4680, and then by Maynard to H 1 ≤ 600, who also established for the first time a finiteness result for H m for m ≥ 2, and specifically that H m m 3 e 4m . If one also assumes the Elliott-Halberstam conjecture, Maynard obtained the bound H 1 ≤ 12, improving upon the previous bound H 1 ≤ 16 of Goldston, Pintz, and Yıldırım, as well as the bound H m m 3 e 2m . In this paper, we extend the methods of Maynard by generalizing the Selberg sieve further and by performing more extensive numerical calculations. As a consequence, we can obtain the bound H 1 ≤ 246 unconditionally and H 1 ≤ 6 under the assumption of the generalized Elliott-Halberstam conjecture. Indeed, under the latter conjecture, we show the stronger statement that for any admissible triple (h 1 , h 2 , h 3 ), there are infinitely many n for which at least two of n + h 1 , n + h 2 , n + h 3 are prime, and also obtain a related disjunction asserting that either the twin prime conjecture holds or the even Goldbach conjecture is asymptotically true if one allows an additive error of at most 2, or both. We also modify the 'parity problem' argument of Selberg to show that the H 1 ≤ 6 bound is the best possible that one can obtain from purely sieve-theoretic considerations. For larger m, we use the distributional results obtained previously by our project to obtain the unconditional asymptotic bound

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Section: Organization Of the Papermentioning

confidence: 99%

Section: Distribution Estimates On Arithmetic Functionsmentioning

confidence: 99%

Variants of the Selberg sieve, and bounded intervals containing many primes

Polymath¹

2014

Res Math Sci

130

172

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“…Prior to [4], a bilinear structure in the error term of the Λ 2 sieve was observed in [10] and the first improvement upon (5.4) was achieved; see [5] as well. Later the development [13] made it possible to prove (5.2) via the Λ 2 sieve; see [17] for a further development. On the other hand, the bound (5.9) depends on our large sieve extension of the Λ 2 sieve that is devised via the duality principle and the quasi-character derived from optimal Λ 2 -weights, as is already mentioned in the first section.…”

Section: Main Theoremmentioning

confidence: 99%

An extension of the Linnik phenomenon

Motohashi

2013

Proc. Steklov Inst. Math.

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Abstract:We shall prove an extension of Yu.V. Linnik's phenomenon concerning C.L. Siegel's exceptional zeros of Dirichlet L-functions. Also we shall prove a new version of the Brun-Titchmarsh theorem. These two subjects are closely related to each other. The basic tool is our theory of A. Selberg's Λ 2 -sieve, which is developed in our old Tata lecture notes [12]. Constants which are involved either explicitly or implicitly in our discussion are all effectively computable, save for the one in the statement of a theorem of Siegel which has no relation with our main assertions. We add that the discussion in Sections 1-4 is a substantially revised and augmented version of Section 9.3 of our recent monograph [15, Vol. I] and that of Section 5 an extraction from our article [16] posted in arXiv.Keywords: Siegel's zeros; Λ 2 -sieve; Brun-Titchmarsh theorem 1. Our motivation in a historical perspective. We shall first make explicit our notion of exceptional zeros: Let χ denote a genetic Dirichlet character, with which the L-function L(s, χ) is associated. Let T > 0 be sufficiently large, and let Z T = {ρ = β + iγ} be the set of all non-trivial zeros ρ in the region |Im s| ≤ T of the function q≤T * If β T exists, it should be real and simple, and we designate both itself and the relevant unique primitive character χ T , with L(β T , χ T ) = 0, as T -exceptional. It is known that χ T is real. As a matter of fact, we have more preciselyThe upper bound is a consequence of the Dirichlet class number formula for quadratic number fields. However, the formula can be dispensed with. See [7, Section 2, Chapter IX] as well as [15, Section 4.5, Vol. I]; in both monographs are comprehensible accounts of the theory of the zeta and Dirichlet L-functions and the theory of the distribution of primes. One of the most tantalising problems in number theory is the elimination of the possibility of the existence of exceptional zeros. It is generally believed that they do not exist at all. A way to confirm this is to improve the Brun-Titchmarsh theorem in the manner to be made explicit at a later part of Section 5, which remains, however, to be one of the most difficult problems in number theory. Also one may have a hope in employing our Theorem 1 for this purpose; we shall try to be precise in one of the remarks there.

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“…First, he isolated a weaker distribution estimate that sufficed to obtain the bounded gap property (still involving the crucial feature of going beyond the range accessible to the BombieriVinogradov technique), where (roughly speaking) only smooth 1 moduli were involved, and the residue classes had to obey strong multiplicative constraints (the possibility of such a weakening had been already noticed by Motohashi and Pintz [43]). Secondly, and more significantly, Zhang then proved such a distribution estimate.…”

Section: Introductionmentioning

confidence: 99%